Linear Systems

What is the resonant frequency of a series RLC circuit?

f_0 = \frac{1}{2\pi\sqrt{LC}}
At resonance, inductive and capacitive reactances cancel and impedance is purely resistive.

What is the relationship between quality factor and bandwidth in a series RLC circuit?

Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 R C}, \quad \text{and} \quad \text{BW} = \frac{\omega_0}{Q}

Higher Q means a narrower bandwidth and sharper resonance.

What is the cutoff frequency of a first-order RC low-pass filter?

f_c = \frac{1}{2\pi RC}
At this frequency, |H(j\omega)| = \frac{1}{\sqrt{2}} of its low-frequency value.

What determines whether an RL circuit acts as a low-pass or high-pass filter?

\omega_c = \frac{R}{L} \quad (fc = \frac{R}{2\pi L})
If output is taken across the resistor → low-pass.
If output is taken across the inductor → high-pass.

Define the transfer function G(s).

G(s) = \frac{Y(s)}{U(s)}
It represents the system response with all initial conditions set to zero and is independent of them.

What determines closed-loop stability in an LTI system?

Stability depends on pole locations.
The system is stable if all poles lie in the left-half of the s-plane.

What is phase margin?

It is how much additional phase lag at the gain-crossover frequency would make the system unstable.
Measured in degrees; higher margin = greater stability.

What is the Laplace transform of a derivative?

\mathcal{L}{f'(t)} = sF(s) - f(0)
\mathcal{L}{f''(t)} = s^2F(s) - s f(0) - f'(0)

What is the Laplace transform of a delayed signal?

x(t - t0)u(t - t0) \;\longleftrightarrow\; e^{-s t_0} X(s) - This represents a shift in the time domain, where the signal is delayed by t_0 seconds.

What is the current response of an RL discharge circuit?

i(t) = I_0 e^{-\frac{R}{L}t}
Time constant \tau = \frac{L}{R}.

Give the standard form of a second-order transfer function and define its parameters.

G(s) = \frac{\omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2} \omega_n = natural frequency; \zeta = damping ratio.

Approximate 2% settling time for a second-order system?

Ts \approx \frac{4}{\zeta \omega_n}

What is steady-state error for a unit ramp input (unity feedback)?

K_v = \lim_{s \to 0} s G(s)

e_{ss} = \frac{1}{K_v}

Practical: Find resonant frequency if L = 20\text{ mH} and C = 100\text{ nF}.

Answer: f_0 = \frac{1}{2\pi\sqrt{(20\times10^{-3})(100\times10^{-9})}} \approx 3.56\text{ kHz}
Steps: Use the resonance formula and plug in L, C.

Practical: Given f_0 = 159.2\text{ Hz} and Q = 2, find bandwidth.

Answer:

\text{BW}\omega = \frac{\omega0}{Q} = \frac{1000}{2} = 500\text{ rad/s}
Convert: \text{BW}_f = \frac{500}{2\pi} \approx 79.6\text{ Hz}

Practical: Find the cutoff frequency of an RC low-pass filter with R = 10\text{ k}\Omega and C = 1\mu\text{F}.

Answer: f_c = \frac{1}{2\pi(10\times10^3)(1\times10^{-6})} \approx 15.9\text{ Hz}

Steps: Use f_c = \frac{1}{2\pi RC}.

Practical: Compute \mathcal{L}{t e^{-2t}}.

Answer: \frac{1}{(s + 2)^2}
Steps: Use property \mathcal{L}{t f(t)} = -\frac{d}{ds}F(s) with F(s) = \frac{1}{s+2}.

What is the first shifting theorem in Laplace transforms?

\mathcal{L}{e^{at} f(t)} = F(s - a)

Practical: Find equivalent transfer function for two blocks in parallel:

G1(s) = \frac{2s}{s+3}, \quad G2(s) = \frac{1}{s^2 + 4s + 5}

Answer:

G{eq}(s) = G1 + G_2 = \frac{2s^3 + 8s^2 + 11s + 3}{(s + 3)(s^2 + 4s + 5)}
Steps: Add using common denominator and combine numerators.

What does adding derivative control K_d s do in a feedback system?

Increases damping ratio \zeta, reduces overshoot, and improves transient response.

What is closed-loop bandwidth?

Frequency range where system magnitude stays within -3 dB of its low-frequency gain.
Wider bandwidth → faster response.

What is the Z-transform time shift property?

x[n - n0] \;\longleftrightarrow\; z^{-n0} X(z)

Practical: Find y[2] for convolution of
x[n] = {1, 2, -1} and h[n] = {2, 1, -1}.

Answer: y[2] = -1
Steps: y[n] = \sum_k x[k]h[n-k]
y[2] = 1(-1) + 2(1) + (-1)(2) = -1

Write the solution form for a first-order homogeneous ODE.

y' + a y = 0 \;\Rightarrow\; y = Ce^{-a t}
Template for RL/RC exponential responses.

What does a Bode magnitude plot show?

It shows system gain in decibels vs frequency.
\text{Magnitude (dB)} = 20\log_{10}|H(j\omega)|
Used to visualize filter or control system response.

What does a Bode phase plot show?

It shows the phase angle of H(j\omega) vs frequency.
\phi(\omega) = \tan^{-1}!\left(\frac{\text{Im}[H(j\omega)]}{\text{Re}[H(j\omega)]}\right)

What is the slope of a first-order low-pass filter beyond its cutoff frequency?

-20\ \mathrm{dB/decade} or -6\ \mathrm{dB/octave}

What is the magnitude at the cutoff frequency for a first-order filter?

|H(j\omega_c)| = \frac{1}{\sqrt{2}} \approx -3\,\text{dB}

What is the phase shift at cutoff for an RC low-pass filter?

-45^\circ

Practical: Determine the slope segments for H(s) = \frac{100}{s(s+10)}.

Answer:
Two poles (0 rad/s and 10 rad/s).
Steps:

1. Start at +40 dB/dec (from constant 100).

2. At 0 rad/s pole → slope −20 dB/dec.

3. At 10 rad/s pole → slope −40 dB/dec.

How can you find steady-state gain from a Bode plot?

Extrapolate the low-frequency magnitude (0 rad/s).
At DC, |H(0)| gives system’s steady-state gain.

What is the inverse Laplace of \frac{1}{s+a}?

e^{-a t}u(t)

What is the inverse Laplace of \frac{s}{s^2 + \omega^2}?

\cos(\omega t)u(t)

What is the inverse Laplace of \frac{\omega}{s^2 + \omega^2}?

\sin(\omega t)u(t)

Practical: Find \mathcal{L}^{-1}\left{\frac{5}{s^2 + 25}\right}.

Answer: \sin(5t)u(t)
Steps: Compare with \frac{\omega}{s^2 + \omega^2} where \omega = 5.

Practical: Find \mathcal{L}^{-1}\left{\frac{s + 3}{s^2 + 4s + 5}\right}.

Answer: e^{-2t}\big(\cos t + \sin t\big)u(t)
Steps:

1) Complete square: s^2 + 4s + 5 = (s + 2)^2 + 1.
2) Shift and use cosine/sine formulas.

Define underdamped, critically damped, and overdamped systems.

\zeta < 1 → underdamped (oscillatory) \zeta = 1 → critical (fastest non-oscillatory) \zeta > 1 → overdamped (slow non-oscillatory)

Describe step response shape for an underdamped 2nd-order system.

Exponentially decaying sinusoid oscillating about final value.

y(t) = 1 - \frac{1}{\sqrt{1-\zeta^2}} e^{-\zeta\omega_n t} \sin(\omega_d t + \phi)

Define \omega_d for an underdamped system.

\omega_d = \omega_n\sqrt{1 - \zeta^2}

Practical: Given \zeta = 0.5 and

\omega_n = 8\text{ rad/s}, find Ts and \omega_d

Answer:

Ts = \frac{4}{\zeta\omega_n} = 1.0\text{ s}, \quad \omega_d = 8\sqrt{1 - 0.25} = 6.93\text{ rad/s}

For unity feedback, what system type eliminates steady-state error for a step input?

Type 1 (one pole at origin).
Type 0 → finite step error, Type 1 → zero step error, Type 2 → zero ramp error.

What is the error constant for a parabolic (acceleration) input?

Ka = \lim{s \to 0} s^2 G(s)

e{ss} = \frac{1}{Ka} for unit-acceleration input.

How do zeros affect transient response?

Adding zeros increases overshoot and speeds up response; adds phase lead.

How do poles near the imaginary axis affect stability?

The closer poles are to the imaginary axis, the slower the response and less stable the system.

What is the relationship between bandwidth and rise time for second-order systems?

Approximately \text{BW} \cdot T_r \approx 0.35 for small damping ratios.

What does “frequency response” mean?

The steady-state output amplitude and phase of an LTI system when driven by a sinusoid of varying frequency.

Practical: For H(s) = \frac{10}{s + 10}, find magnitude and phase at \omega = 10\,\text{rad/s}.

Answer: |H(j10)| = \frac{10}{\sqrt{10^2 + 10^2}} = 0.707,\; \phi = -45^\circ

Practical: A system has poles at (s=-2) and (s=-5). Find the time constants.

Answer:

\tau1 = \frac{1}{2} = 0.5\,\text{s}, \quad \tau2 = \frac{1}{5} = 0.2\,\text{s}

What is the initial and final value theorem in Laplace transforms?

Initial: f(0^+) = \lim_{s \to \infty} sF(s)

Final: f(\infty) = \lim_{s \to 0} sF(s) (if poles in LHP)

Practical: Use the Final Value Theorem on F(s) = \frac{10}{s(s + 5)}.

Answer: \lim_{s \to 0} sF(s) = \frac{10}{5} = 2
So final value is 2.

Explain what a pole-zero plot represents.

It shows system poles (X) and zeros (O) in the complex s-plane.
Pole locations determine stability; zero locations affect shape of response.